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Heat and Mass Transfer
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MODULE 4
THERMAL RADIATION
Thermal radiation is the electromagnetic radiation emitted by a body as a result of its
temperature. It does not require any material medium for propagation and one uses the attributes
of wavelength or frequency to describe it. The intensity of such energy flux depends upon the
temperature of the body and the nature of its surface.
BLACKBODY RADIATION
A blackbody is defined as a perfect emitter and absorber of radiation. At a specified
temperature and wavelength, no surface can emit more energy than a blackbody. A blackbody
absorbs all incident radiation, regardless of wavelength and direction. Also, a blackbody emits
radiation energy uniformly in all direction per unit area normal to direction of emission. That is,
a blackbody is a diffuse emitter. The term diffuse means independent of direction.
Fig.4.1 A blackbody is said to be a diffuse emitter since it emits radiation energy uniformly
in all directions
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STEFANBOLTZMANN LAW
The radiation energy emitted by a blackbody per unit time and per unit surface area can be
expressed as
(1)
where = 5.67 x 10-8
W/m2K
4 is the StefanBoltzmann constant and T is the absolute
temperature of the surface in K. Eq.1 is known as the StefanBoltzmann law and Eb is called
the blackbody emissive power. The emission of thermal radiation is proportional to the fourth
power of the absolute temperature.
SPECTRAL BLACKBODY EMISSIVE POWER
The StefanBoltzmann law in Eq. 1 gives the total blackbody emissive power Eb, which is the
sum of the radiation emitted over all wavelengths. Spectral blackbody emissive power is the
amount of radiation energy emitted by a blackbody at an absolute temperature T per unit time,
per unit surface area, and per unit wavelength about the wavelength . The relation for the
spectral blackbody emissive power Eb is expressed by Plancks law given as,
(2)
Also, T is the absolute temperature of the surface, is the wavelength of the radiation emitted,
and k =1.38065 X 10-23
J/K is Boltzmanns constant.
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Fig.4.2. The variation of the blackbody emissive power with wavelength for several
temperatures.
WIENS DISPLACEMENT LAW
From Fig.4.2 we can see that as the temperature increases, the peak of the curve shifts toward
shorter wavelengths. The wavelength at which the peak occurs for a specified temperature is
given by Wiens displacement law as,
(3)
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BLACKBODY RADIATION FUNCTION
Black body radiation function f represents the fraction of radiation emitted from a blackbody at
temperature T in the wavelength band from = 0 to . The values of f are tabulated as a function
of T, where is in m and T is in K.
(4)
The fraction of radiation energy emitted by a blackbody at temperature T over a finite
wavelength band from =1 to =2 is determined from,
(5)
Where f1(T ) and f2(T ) are blackbody radiation functions corresponding to 1T and 2T,
respectively.
Fig.4.3 Graphical representation of the fraction of radiation emitted in the wavelength
band from 1 to 2
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THE TOTAL HEMISPHERICAL EMISSIVITY (OR SIMPLY, EMISSIVITY)
The emissivity of a surface represents the ratio of the radiation emitted by the surface at a given
temperature to the radiation emitted by a blackbody at the same temperature. The emissivity of a
surface is denoted by , and it varies between zero and one, 0 1. Emissivity is a measure of
how closely a surface approximates a blackbody, for which = 1.
(6)
MONOCHROMATIC HEMISPHERICAL EMISSIVITY
The monochromatic hemispherical emissivity of a surface is the ratio of the monochromatic
hemispherical emissive power of the surface to the monochromatic hemispherical emissive
power of a black surface at the same temperature and wave length.
=
(7)
GRAY SURFACE
A gray surface is a surface having the same value of the monochromatic (spectral) hemispherical
emissivity at all wave lengths.
ABSORPTIVITY, REFLECTIVITY, AND TRANSMISSIVITY
When radiation strikes a surface, part of it is absorbed, part of it is reflected, and the remaining
part, if any, is transmitted, as illustrated in Fig.4.4. The fraction of irradiation (radiation flux
incident on a surface) absorbed by the surface is called the absorptivity , the fraction reflected
by the surface is called the reflectivity , and the fraction transmitted is called the
transmissivity . That is,
(8)
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(9)
(10)
where G is the radiation energy incident on the surface, and Gabs, Gref, and Gtr are the absorbed,
reflected, and transmitted portions of it, respectively.
Also, (11)
Fig.4.4.The absorption, reflection, and transmission of incident radiation by a
semitransparent material
SPECULAR AND DIFFUSE REFLECTION
In practice, surfaces are assumed to reflect radiation in a perfectly specular or diffuse manner. In
specular (or mirror like) reflection, the angle of reflection equals the angle of incidence of the
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Fig.4.5 Different types of reflection from a surface: (a) actual or irregular, (b) diffuse, and
(c) specular or mirror like.
radiation beam. In diffuse reflection, radiation is reflected equally in all directions, as shown in
Fig.4.5. Reflection from smooth and polished surfaces approximates specular reflection, whereas
reflection from rough surfaces approximates diffuse reflection. In radiation analysis, smoothness
is defined relative to wavelength. A surface is said to be smooth if the height of the surface
roughness is much smaller than the wavelength of the incident radiation.
KIRCHOFFS LAW
Fig 4.6 Sketch showing model for deriving Kirchoffs law.
Consider a perfectly black enclosure as shown in fig.4.6.The surface of the enclosure absorbs all
the incident radiation falling upon it. This enclosure will also emit radiation according to the T4
law. Let the radiant flux arriving at some area in the enclosure be qi W/m2. Now suppose that a
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body is placed inside the enclosure and allowed to come into temperature equilibrium with it. At
equilibrium the energy absorbed by the body must be equal to the energy emitted. At equilibrium
we may write
EA=qiA (12)
If we now replace the body in the enclosure with a blackbody of the same size and shape and
allow it to come to equilibrium with the enclosure at the same temperature,
EbA=qiA (13)
If Equation (12) is divided by Equation (13),
(14)
That is at thermal equilibrium, the ratio of the emissive power of a body to the emissive power of
a blackbody is equal to the absorptivity of the body.
But we know that, the ratio
is defined as the emissivity of the body,
(15)
From equations 14 and 15,
= (16)
That is at thermal equilibrium, the monochromatic emissivity of a surface is equal to its
monochromatic absorptivity. This relation is known as the Kirchoffs law.
THE GREENHOUSE EFFECT
When we leave a car under direct sunlight on a sunny day, the interior of the car gets much
warmer than the air outside, and acts like a heat trap. This can be explained from the spectral
transmissivity curve of the glass, which resembles an inverted U, as shown in Fig.4.7. We
observe from this figure that glass at thicknesses encountered in practice transmits over 90
percent of radiation in the visible range and is practically opaque (nontransparent) to radiation in
the longer-wavelength infrared regions of the electromagnetic spectrum (roughly > 3 m).
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Therefore, glass has a transparent window in the wavelength range 0.3 m < < 3 m in which
Fig.4.7.The spectral transmissivity of low-iron glass at room temperature for different
thicknesses
Over 90 percent of solar radiation is emitted. On the other hand, the entire radiation emitted by
surfaces at room temperature falls in the infrared region. Consequently, glass allows the solar
radiation to enter but does not allow the infrared radiation from the interior surfaces to escape.
This causes a rise in the interior temperature as a result of the energy build-up in the car. This
heating effect, which is due to the non gray characteristic of glass (or clear plastics), is known as
the greenhouse effect.
THE VIEW FACTOR
The view factor (shape factor, configuration factor, or angle factor) from a surface i to a surface j
is denoted by Fij or just Fij, and is defined as the fraction of the radiation leaving surface i that
strikes surface j directly.
RECIPROCITY RELATION
The reciprocity relation for view factors is given by,
A1F12=A2F21 (17)
Where A1 and A2 are the area of the surfaces 1 and 2 respectively.
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THE SUMMATION RULE
The sum of the view factors from surface i of an enclosure to all surfaces of the enclosure,
including to itself, must equal unity. This is known as the summation rule for an enclosure and
is expressed as
(18)
where N is the number of surfaces of the enclosure. For example, applying the summation rule to
surface 1 of a three-surface enclosure yields
(19)
THE SUPERPOSITION RULE
Sometimes the view factor associated with a given geometry is not available in standard tables
and charts. In such cases, it is desirable to express the given geometry as the sum or difference of
some geometries with known view factors, and then to apply the superposition rule, which can
be expressed as the view factor from a surface i to a surface j is equal to the sum of the view
factors from surface i to the parts of surface j.. Consider the geometry in Figure, which is
infinitely long in the direction perpendicular to the plane of the paper. The radiation that leaves
surface 1 and strikes the combined surfaces 2 and 3 is equal to the sum of the radiation that
strikes surfaces 2 and 3. Therefore, the view factor from surface 1 to the combined surfaces of 2
and 3 is,
F 1(2, 3) = F 1 2 + F 13 (20)
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Fig.4.8 The view factor from a surface to a composite surface is equal to the sum of the
view factors from the surface to the parts of the composite surface.
RADIOSITY
Surfaces emit radiation as well as reflect it, and thus the radiation leaving a surface consists of
emitted and reflected parts. The calculation of radiation heat transfer between surfaces involves
the total radiation energy streaming away from a surface, with no regard for its origin. The total
radiation energy leaving a surface per unit time and per unit area is the radiosity and is denoted
by J. For a surface i radiosity can be expressed as,
Ji= (Radiation emitted by surface i) + (Radiation reflected by surface i)
RADIATION SHIELDS
Radiation heat transfer between two surfaces can be reduced greatly by inserting a thin, high-
reflectivity (low-emissivity) sheet of material between the two surfaces. Such highly reflective
thin plates or shells are called radiation shields. Multilayer radiation shields constructed of
about 20 sheets per cm thickness separated by evacuated space are commonly used in cryogenic
and space applications. The role of the radiation shield is to reduce the rate of radiation heat
transfer by placing additional resistances in the path of radiation heat flow. The lower the
emissivity of the shield, the higher the resistance.
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MASS TRANSFER
Mass transfer requires the presence of two regions at different chemical compositions, and refers
to the movement of a chemical species from a high concentration region toward a lower
concentration one relative to the other chemical species present in the medium. The primary
driving force for fluid flow is the pressure difference, whereas for mass transfer it is the
concentration difference. Therefore, we do not speak of mass transfer in a homogeneous
medium.
FICKS LAW OF DIFFUSION
According to Ficks law of diffusion, the rate of mass diffusion mdiff of a chemical species A in
a stationary medium in the direction x is proportional to the concentration gradient dC/dx in that
direction and is expressed as,
(21)
where DAB is the diffusion coefficient (or mass diffusivity) of the species in the mixture and CA is
the concentration of the species in the mixture at that location.
DIMENSIONLESS PARAMETERS IN CONVECTIVE MASS TRANSFER
SCHMIDT NUMBER:-
.. (22)
which represents the relative magnitudes of molecular momentum and mass diffusion in the
velocity and concentration boundary layers, respectively. The role of Schmidt number is
analogous to role of Prandtle number in convection heat transfer. A Schmidt number of near
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unity (Sc= 1) indicates that momentum and mass transfer by diffusion are comparable, and
velocity and concentration boundary layers almost coincide with each other.
LEWIS NUMBER:-
. (23)
It is the ratio of thermal diffusivity to mass diffusivity. It compares the relative thicknesses of
thermal and concentration boundary layers.
SHERWOOD NUMBER
(24)
It is analogous to Nusselt number in convection heat transfer and it may be expressed as ratio of
concentration gradient at the surface to overall concentration gradient.
ANALOGY BETWEEN HEAT AND MASS TRANSFER 1) The driving force for heat transfer is the temperature difference. In contrast, the driving force
for mass transfer is the concentration difference.
2) The rate of heat conduction in a direction x is proportional to the temperature gradient dT/dx
in that direction and is expressed by Fouriers law of heat conduction as,
.. (25)
where k is the thermal conductivity of the medium and A is the area normal to the direction of
heat transfer. Likewise, the rate of mass diffusion of a chemical species A in a stationary
medium in the direction x is proportional to the concentration gradient dC/dx in that direction
and is expressed by Ficks law of diffusion by
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.. (26)
where DAB is the diffusion coefficient (or mass diffusivity) of the species in the mixture and CA
is the concentration of the species in the mixture at that location.
3) The rate of heat convection for external flow was expressed by Newtons law of cooling as,
.(27)
where hconv is the heat transfer coefficient, As is the surface area, and Ts-T is the temperature
difference across the thermal boundary layer. Likewise, the rate of mass convection can be
expressed as,
.. (28)
where hmass is the mass transfer coefficient, As is the surface area, and Cs C is concentration
difference across the concentration boundary layer.
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